The influence of disturbances in iterative learning control

For systems that perform repetitive tasks, a high performance feedforward signal can be derived using Iterative Learning Control (ILC). The feedforward signal is updated through successive iterations. Disturbances present in the control scheme, such as load and measurement disturbances, are also present in the learning process and deteriorate the performance of ILC. This paper presents an expression for the tracking error of an arbitrary iteration which shows the influence of measurement and load disturbances of the present and previous iterations. The expression is validated by means of simulations and experiments on a motion system. The disturbances of the last two iterations prove to have the largest influence on the tracking error

Modeling of a walking piezo actuator

Piezoelectric actuators are often used in positioning devices that require (sub)nano-meter resolution. In this paper, wedevelop an electro-mechanical dynamic model of a walking piezo actuator. The derived model structure can be used forthe dynamic modeling of bimorph piezo motors in general. Furthermore, the physical nature enables the model to beused in design optimizations to derive new motors with di erent properties and for a dynamic analysis to investigatethe maximum allowable driving frequency in relation to the dynamic e ects of the motor. The walking piezo actuatorcontains four legs, each with two electrically separated piezo stacks. The legs are modeled as a connection of coupledmass-spring-damper systems. Using a Lagrange approach, the nonlinear system dynamics are derived. The variationin the system dynamics is assessed using linearization around di erent equilibrium positions. Also a static linearizedapproximation is derived, which describes the static relation between the supply voltages and the tip trajectories ofthe legs. The dynamic analysis shows that the motor can be modeled su ciently accurate using a connection of sixlumped mass-spring-damper systems. The variation in system dynamics appears to be most signi cant in the movementperpendicular to the leg orientation. Experiments show that the static linearized model accurately describes the tiptrajectories of the legs for both sinusoidal and asymmetric waveforms

Gain scheduling control of a walking piezo actuator

Piezoelectric actuators are commonly used to drive high-precision stages. In this paper, a walking piezo actuator containing four piezoelectric legs is used to drive a high-precision nanomotion stage with constant velocity and point-to-point reference signals. The gain of the system is dependent on the momentary orientation of the piezo legs during the walking movement. The aim of this paper is to design a feedback controller that employs knowledge of the varying gain of the drive legs over one drive cycle. The gain variation is determined by a combination of local and global identification and modeling techniques. The excitation signal for the local frequency response function measurements at different positions of the legs in the drive cycle is designed to avoid stick--slip effects between the drive legs and the drive surface of the stage. Using the knowledge of the system variations, a linear parameter-varying model and a gain scheduling feedback controller are designed for the nanomotion stage. Experiments show that the designed gain scheduling controller reduces the tracking error and settling time compared to a robust ${cal H}_infty$ controller with a comparable closed-loop bandwidth up to 53% and 80%, respectively

Directional repetitive control of a metrological AFM

Atomic force microscopes (AFMs) are used forsample imaging and characterization at nanometer scale. In thiswork, we consider a metrological AFM, which is used for thecalibration of transfer standards for commercial AFMs. Themetrological AFM uses a three degree-of-freedom (DOF) stageto move the sample with respect to the probe of the AFM. Therepetitive sample topography introduces repetitive disturbancesin the system. To suppress these disturbances, repetitive control(RC) is applied to the imaging axis. A rotated sample orientationwith respect to the actuation axes introduces a non-repetitivenessin the originally fully repetitive errors and yields a deterioratedperformance of RC. Directional repetitive control (DRC) isintroduced to align the axes of the scanning movement with thesample orientation under the microscope. Experiments show thatthe proposed directional repetitive controller significantly reducesthe tracking error as compared to standard repetitive control

Delay-varying repetitive control with application to a walking piezo

The performance of systems that exhibit repetitive disturbances can be significantly improved using repetitive control. If the repetitive disturbance is periodic with respect to time, perfect asymptotic disturbance rejection can be achieved by well-known methods. However, many systems have a repetitive nature with respect to a variable other than time. For this type of systems, we propose a delay-varying repetitive control (DVRC) method, which employs a time-varying delay in the repetitive controller that is continuously adjusted based on the repetitive variable. An norm-based criterion is derived that guarantees stability of the time-varying delay system for a given range of variations of the repetitive delay. The strengths of this new repetitive control scheme are shown by applying it to a nanomotion stage driven by a walking piezo actuator

Modeling, identification and control of a metrological Atomic Force Microscope with a 3DOF stage

Atomic Force Microscopes (AFMs) are widely used for the investigation of samples at nanometer scale. In this paper, we present the modeling, the identification and the control of a metrological AFM. The metrological AFM is used for the calibration of transfer standards for commercial AFMs. Therefore, the focus of the presented work is on scanning accuracy rather than on scanning speed. The contribution of this paper is the combination of 3 degree-of-freedom (DOF) control, including position feedforward, with an AFM with fixed cantilever and a piezo-stack driven stage. The amount of coupling between all DOFs is assessed by a non-parametric MIMO identification of the AFM. Since the dynamics appear to be decoupled in the frequency range of interest, feedback controllers are designed using loopshaping techniques for each DOF separately. Position feedforward is added to the stage in x and y direction, which improves the tracking performance by a factor two. The controlled stage is able to track scanning profiles within the sensor bound of 5 nm. With the proposed control method, the metrological AFM can produce images of the transfer standards with a sensor bound of 2 nm. Furthermore, real-time imaging of the sample is possible without the need for a-posteriori image correction. Finally, it is shown that the proposed control method almost completely compensates the hysteresis in the system

Modeling and compensation of asymmetric hysteresis in a piezo actuated metrological AFM

The manipulation of samples in atomic force microscopes (AFMs) is often performed using piezoelectric actuators. In this paper, a metrological AFM with a 3 degree-of-freedom (DOF) stage driven by piezo-stack actuators is considered. The piezo actuators exhibit hysteresis, which can change the system dynamics and/or acts as a non-linear disturbance on the system. This deteriorates the performance of the AFM. The 3 DOF stage exhibits asymmetric hysteresis, which is modeled by extending the Coleman-Hodgdon model. The asymmetry includes a scan range dependent offset and an asymmetry between the trace and retrace directions. Non-linear multi-variable optimization is employed to derive the optimal generic model for all scan ranges. The proposed extended Coleman-Hodgdon model describes the asymmetric hysteresis over all scan ranges with an accuracy of 97%. Based on the model, a feedforward compensation method is developed. Experiments on the metrological AFM show that the application of the hysteresis feedforward largely improves the scanning accuracy